From: mclaren
Subject: Wilson CPS scales - post 3 of 9
As has long since been demonstrated, most of the subscribers to this forum
have no interest in music. To all but a handful of you, music is an annoying
and deeply distasteful waste product of the really IMPORTANT aesthetic
process-- namely, theorizing.
This is a perverse habit of thought peculiar to the late 20th century.
Subsequent eras will look back on ours with wonder and disgust, shaking
their heads. Thus, rather that wander deeper into the tarpit of sterile
theory, here's a list of Wilson CPSs--a list primarily useful to those
of you who actually wish to produce MUSIC, rather than endless mental-masturbation
theory so beloved of Cage/Forte/Babbitt/Boros/Ferneyhough et al.:
First, a Wilson 5 of 10 CPS. 5 out of 1st 10 primes: [1,3,5,7,11,13,17,19,23,29].
All generators are relatively prime and thus this CPS yields the full 252
pitches. Some adjacent scale degrees are not audibly distinct, others are.
Notice that because of their proximity, subsets of the pitches of this
tuning would make excellent Csound/US-type additive synthesis timbres.
In order to perform the entire Wilson scale, This CPS could be split
across 3 MIDI channels and performed live by 3 players on different channels.
Alternatively, it could easily be played with a MIDI theremin by using
a real-time device to switch every other note between channels 1,2,3,1,2,3...
As it happens, I have such a device--the Oberheim Navigator. A cheap little
forty-dollar MIDI box that has obvious value in playing such a scale in
live performance:
1: 65569/65536 0.871526
2: 8211/8192 4.010665
3: 16445/16384 6.433668
4: 8265/8192 15.35893
5: 1062347/1048576 22.58838
6: 266133/262144 26.14547
7: 33495/32768 37.98977
8: 134589/131072 45.84119
9: 33649/32768 45.93123
10: 1077205/1048576 46.63370
11: 33915/32768 59.56307
12: 34017/32768 64.76198
13: 17017/16384 65.62695
14: 17043/16384 68.27005
15: 2145/2048 80.11435
16: 551551/524288 87.76175
17: 17255/16384 89.67227
18: 69069/65536 90.90090
19: 8645/8192 93.18034
20: 138567/131072 96.26907
21: 139403/131072 106.6825
22: 279565/262144 111.3891
23: 70035/65536 114.9462
24: 4389/4096 119.6119
25: 140505/131072 120.3143
26: 70499/65536 126.3782
27: 8855/8192 134.7319
28: 17765/16384 140.1001
29: 35581/32768 142.5833
30: 572033/524288 150.8867
31: 286143/262144 151.6522
32: 35815/32768 153.9316
33: 4485/4096 157.0708
34: 1153243/1048576 164.7182
35: 72105/65536 165.3740
36: 289731/262144 173.2255
37: 145145/131072 176.5625
38: 18183/16384 180.3632
39: 36465/32768 185.0698
40: 36685/32768 195.4832
41: 9177/8192 196.5683
42: 146965/131072 198.1358
43: 147407/131072 203.3347
44: 1155/1024 208.4126
45: 2369851/2097152 211.6380
46: 4641/4096 216.2640
47: 37145/32768 217.0565
48: 74613/65536 224.5673
49: 150423/131072 238.3988
50: 150535/131072 239.6874
51: 303485/262144 253.5189
52: 37961/32768 254.6765
53: 38019/32768 257.3196
54: 19019/16384 258.1846
55: 608855/524288 258.8871
56: 76245/65536 262.0262
57: 4785/4096 269.1639
58: 19227/16384 277.0153
59: 154077/131072 279.9505
60: 19285/16384 282.2299
61: 309111/262144 285.3186
62: 2415/2048 285.3690
63: 4845/4096 290.7373
64: 623645/524288 300.4387
65: 156009/131072 301.5238
66: 312455/262144 303.9468
67: 19635/16384 313.3681
68: 78793/65536 318.9359
69: 9867/8192 322.0751
70: 39585/32768 327.1996
71: 79373/65536 331.6329
72: 39767/32768 335.1410
73: 10005/8192 346.1204
74: 5005/4096 346.9853
75: 646323/524288 362.2751
76: 323323/262144 363.1401
77: 81345/65536 374.1194
78: 40755/32768 377.6275
79: 81719/65536 382.0609
80: 327845/262144 387.1854
81: 41055/32768 390.3245
82: 164749/131072 395.8923
83: 10353/8192 405.3136
84: 20735/16384 407.7366
85: 5187/4096 408.8217
86: 667667/524288 418.5232
87: 1339481/1048576 423.8914
88: 10465/8192 423.9418
89: 167739/131072 427.0305
90: 20995/16384 429.3100
91: 168245/131072 432.2451
92: 676039/524288 440.0965
93: 42427/32768 447.2342
94: 5313/4096 450.3734
95: 170085/131072 451.0758
96: 85085/65536 451.9408
97: 85215/65536 454.5839
98: 10659/8192 455.7415
99: 21489/16384 469.5730
100: 21505/16384 470.8616
101: 43355/32768 484.6931
102: 87087/65536 492.2039
103: 697015/524288 492.9964
104: 1365/1024 497.6225
105: 2800733/2097152 500.8478
106: 21945/16384 505.9258
107: 22011/16384 511.1247
108: 352495/262144 512.6921
109: 88179/65536 513.7772
110: 88711/65536 524.1907
111: 177905/131072 528.8972
112: 22287/16384 532.6979
113: 11165/8192 536.0349
114: 44863/32768 543.8864
115: 2805/2048 544.5422
116: 721259/524288 552.1896
117: 90321/65536 555.3288
118: 11305/8192 557.6082
119: 5655/4096 558.3738
120: 90915/65536 566.6771
121: 182091/131072 569.1603
122: 365313/262144 574.5285
123: 45885/32768 582.8822
124: 23023/16384 588.9461
125: 737035/524288 589.6486
126: 46189/32768 594.3143
127: 11571/8192 597.8713
128: 23205/16384 602.5780
129: 23345/16384 612.9914
130: 374187/262144 616.0802
131: 46835/32768 618.3596
132: 187473/131072 619.5882
133: 5865/4096 621.4987
134: 1508087/1048576 629.1461
135: 189805/131072 640.9904
136: 190095/131072 643.6336
137: 95095/65536 644.4985
138: 95381/65536 649.6974
139: 3003/2048 662.6268
140: 96135/65536 663.3292
141: 24035/16384 663.4193
142: 96577/65536 671.2707
143: 12155/8192 683.1150
144: 3045/2048 686.6721
145: 391391/262144 693.9015
146: 196707/131072 702.8268
147: 393965/262144 705.2498
148: 49335/32768 708.3890
149: 396865/262144 717.9469
150: 198835/131072 721.4550
151: 24871/16384 722.6126
152: 12441/8192 723.3781
153: 50141/32768 736.4441
154: 3135/2048 737.1000
155: 6279/4096 739.5833
156: 12597/8192 744.9514
157: 100947/65536 747.8865
158: 1621477/1048576 754.6528
159: 25415/16384 760.0714
160: 51051/32768 767.5822
161: 408595/262144 768.3748
162: 51359/32768 777.9957
163: 823745/524288 782.2062
164: 103037/65536 783.3639
165: 12903/8192 786.5030
166: 51765/32768 791.6275
167: 25935/16384 795.1356
168: 52003/32768 799.5690
169: 26013/16384 800.3345
170: 418209/262144 808.6378
171: 6545/4096 811.4133
172: 6555/4096 814.0564
173: 13195/8192 825.2448
174: 211497/131072 828.3335
175: 212135/131072 833.5482
176: 3315/2048 833.7521
177: 26565/16384 836.6873
178: 852397/524288 841.3995
179: 53295/32768 842.0554
180: 106743/65536 844.5387
181: 6699/4096 851.6764
182: 107445/65536 855.8870
183: 215441/131072 860.3203
184: 27115/16384 872.1646
185: 6783/4096 873.2497
186: 13585/8192 875.6727
187: 873103/524288 882.9512
188: 437437/262144 886.4592
189: 13685/8192 888.3698
190: 110055/65536 897.4385
191: 442221/262144 905.2900
192: 443555/262144 910.5045
193: 111435/65536 919.0119
194: 55913/32768 925.0758
195: 14007/8192 928.6328
196: 224315/131072 930.2003
197: 28101/16384 934.0010
198: 1812239/1048576 947.2105
199: 56695/32768 949.1210
200: 28405/16384 952.6291
201: 113883/65536 956.6318
202: 57057/32768 960.1400
203: 1785/1024 962.0505
204: 115115/65536 975.2600
205: 14421/8192 979.0607
206: 230945/131072 980.6282
207: 57855/32768 984.1853
208: 29029/16384 990.2492
209: 7293/4096 998.7565
210: 7315/4096 1003.971
211: 29393/16384 1011.822
212: 236379/131072 1020.891
213: 7395/4096 1022.801
214: 3705/2048 1026.309
215: 238119/131072 1033.588
216: 476905/262144 1036.011
217: 119301/65536 1037.096
218: 15015/8192 1048.940
219: 30107/16384 1053.374
220: 482885/262144 1057.584
221: 30305/16384 1064.722
222: 60697/32768 1067.205
223: 3795/2048 1067.861
224: 121771/65536 1072.573
225: 975821/524288 1075.508
226: 15249/8192 1075.712
227: 15295/8192 1080.927
228: 245157/131072 1084.016
229: 493493/262144 1095.204
230: 494247/262144 1097.847
231: 7735/4096 1100.623
232: 124355/65536 1108.926
233: 62205/32768 1109.692
234: 124729/65536 1114.125
235: 62491/32768 1117.633
236: 250705/131072 1122.757
237: 31395/16384 1125.897
238: 3927/2048 1127.054
239: 62985/32768 1131.265
240: 7917/4096 1140.886
241: 63365/32768 1141.678
242: 127281/65536 1149.189
243: 1995/1024 1154.608
244: 32045/16384 1161.374
245: 256795/131072 1164.309
246: 515185/262144 1169.677
247: 1031849/524288 1172.161
248: 64515/32768 1172.817
249: 260015/131072 1185.883
250: 130065/65536 1186.648
251: 16269/8192 1187.806
252: 8151/4096 1191.314
Next, a scale which conveniently fits into the 127 slots
of MIDI. Wilson 3 out of 10 CPS. 3 out of the set of 10 generators [1,3,5,7,11,13,17,19,23,29]
(first 10 primes). As a rule of thumb, for n out of m generators when n
is close to m/2 the result is a pitch set of maximum size. The farther
n is from m/2 (either > m/2 or < m/2), the smaller the resulting pitch
set.
1: 4147/4096 21.42280
2: 65/64 26.84138
3: 1045/1024 35.14468
4: 2093/2048 37.62793
5: 4199/4096 42.99610
6: 33/32 53.27296
7: 133/128 66.33895
8: 4301/4096 84.54773
9: 8671/8192 98.37924
10: 273/256 111.3086
11: 2185/2048 112.1011
12: 551/512 127.0902
13: 69/64 130.2294
14: 1105/1024 131.7968
15: 2233/2048 149.7211
16: 35/32 155.1396
17: 561/512 158.2284
18: 2261/2048 171.2944
19: 1131/1024 172.0599
20: 285/256 185.7818
21: 143/128 191.8457
22: 145/128 215.8910
23: 4669/4096 226.6775
24: 9367/8192 232.0457
25: 1173/1024 235.1848
26: 595/512 260.0951
27: 299/256 268.8021
28: 4807/4096 277.1054
29: 2431/2048 296.8011
30: 609/512 300.3582
31: 77/64 320.1440
32: 2465/2048 320.8464
33: 1235/1024 324.3545
34: 39/32 342.4828
35: 627/512 350.7861
36: 1265/1024 365.9061
37: 5083/4096 373.7575
38: 319/256 380.8953
39: 161/128 397.1004
40: 323/256 402.4685
41: 1309/1024 425.0994
42: 1311/1024 427.7425
43: 2639/2048 438.9309
44: 165/128 439.5868
45: 663/512 447.4382
46: 665/512 452.6528
47: 667/512 457.8517
48: 21/16 470.7811
49: 5423/4096 485.8507
50: 2717/2048 489.3588
51: 85/64 491.2693
52: 2737/2048 502.0559
53: 2755/2048 513.4041
54: 345/256 516.5433
55: 87/64 531.5324
56: 11339/8192 562.8071
57: 5681/4096 566.3152
58: 357/256 575.7365
59: 715/512 578.1596
60: 91/64 609.3538
61: 1463/1024 617.6571
62: 1479/1024 636.4879
63: 741/512 639.9959
64: 1495/1024 655.1160
65: 187/128 656.2736
66: 377/256 670.1051
67: 6061/4096 678.4084
68: 759/512 681.5476
69: 95/64 683.8270
70: 3059/2048 694.6135
71: 385/256 706.4579
72: 1547/1024 714.3093
73: 195/128 728.7967
74: 391/256 733.2301
75: 12673/8192 755.3649
76: 1595/1024 767.2092
77: 399/256 768.2943
78: 6409/4096 775.0606
79: 805/512 783.4143
80: 1615/1024 788.7825
81: 203/128 798.4034
82: 51/32 806.9107
83: 3289/2048 820.1203
84: 1653/1024 829.0455
85: 3335/2048 844.1657
86: 209/128 848.8313
87: 105/64 857.0950
88: 429/256 893.8010
89: 3451/2048 903.3589
90: 1729/1024 906.8670
91: 435/256 917.8463
92: 437/256 925.7877
93: 55/32 937.6320
94: 221/128 945.4835
95: 1771/1024 948.4186
96: 3553/2048 953.7868
97: 7163/4096 967.6183
98: 897/512 970.7575
99: 455/256 995.6677
100: 57/32 999.4685
101: 7337/4096 1009.169
102: 115/64 1014.588
103: 231/128 1022.099
104: 7429/4096 1030.743
105: 935/512 1042.587
106: 1885/1024 1056.419
107: 119/64 1073.781
108: 957/512 1082.850
109: 15/8 1088.269
110: 3857/2048 1095.916
111: 483/256 1099.055
112: 969/512 1104.423
113: 1955/1024 1119.543
114: 493/256 1134.533
115: 247/128 1138.041
116: 2001/1024 1159.807
117: 1001/512 1160.672
118: 253/128 1179.592
119: 1015/512 1184.717
120: 255/128 1193.224
Next posting, a continuation of this garland of Wilson CPSs which boldly
goes where no xenharmonist has gone before...
--mclaren
